Subgroups — #26114997

Order 12032.

322 conjugacy classes of subgroups.

The subgroup lattice has 218 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20243560	C₂	1	1	yes	maximal, characteristic
2	#20242162	C₂	1	2	yes	maximal
3	#20242136	C₂	1	1	yes	maximal, characteristic
4	#20241291	C₂	1	2	yes	maximal
5	C₈×C₄⋊C₄×C₄₇	C₂	1	1	yes	maximal, characteristic
6	#15689176	C₂²	1	2	yes	—
7	C₂×#13093960	—	2	1	no	—
8	C₂×#13093960	—	2	1	no	—
9	#15689147	C₂²	1	2	yes	—
10	#15688876	C₄	1	2	yes	complemented
11	#15688875	—	2	4	no	—
12	#15688874	C₄	1	2	yes	complemented
13	C₂×C₄⋊C₈×C₄₇	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₈×C₄₇	—	2	1	no	—
15	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
16	C₄×C₄⋊C₄×C₄₇	C₂²	1	1	yes	characteristic
17	C₄₇×#336	—	2	1	no	—
18	C₂×#10625329	—	2	2	no	—
19	#13093960	—	4	2	no	—
20	#13093960	—	4	2	no	—
21	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
22	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
23	C₄⋊(C₄₇⋊C₈)	C₂×C₄	1	4	yes	—
24	C₂²⋊(C₄₇⋊C₈)	—	2	2	no	—
25	C₄×C₄₇⋊C₈	—	2	2	no	—
26	C₂²×C₈×C₄₇	—	2	1	no	—
27	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
28	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
29	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
30	C₂×C₄²×C₄₇	—	2	1	no	—
31	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
32	C₄⋊C₈×C₄₇	—	2	1	no	—
33	C₄⋊C₈×C₄₇	D₄	1	2	yes	—
34	C₄×C₈×C₄₇	—	4	2	no	—
35	C₄×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
36	C₄×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
37	C₄×C₈×C₄₇	Q₈	1	1	yes	characteristic
38	C₄×C₈×C₄₇	Q₈	1	1	yes	characteristic
39	#95×C₄₇	Q₈	1	2	yes	—
40	C₂²×C₄×C₄₇	—	2	2	no	—
41	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
42	C₂²×C₄×C₄₇	C₂×Q₈	1	1	yes	characteristic
43	C₂²×C₄×C₄₇	#41	1	1	yes	characteristic
44	C₂×C₈×C₄₇	—	2	1	no	—
45	C₂×C₈×C₄₇	—	2	1	no	—
46	C₂×C₈×C₄₇	—	2	2	no	—
47	C₂×C₈×C₄₇	—	4	1	no	—
48	C₂×C₈×C₄₇	—	4	1	no	—
49	C₂×C₈×C₄₇	C₄⋊C₄	1	1	yes	characteristic
50	C₂×C₈×C₄₇	C₄⋊C₄	1	1	yes	characteristic
51	C₂×C₈×C₄₇	C₄⋊C₄	1	1	yes	characteristic
52	C₂×C₈×C₄₇	C₄⋊C₄	1	1	yes	characteristic
53	C₂×C₈×C₄₇	D₈	1	2	yes	—
54	C₂×C₈×C₄₇	Q₁₆	1	2	yes	—
55	C₄⋊C₄×C₄₇	—	2	4	no	—
56	C₄²×C₄₇	—	4	1	no	—
57	C₄²×C₄₇	—	4	1	no	—
58	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
59	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
60	C₄²×C₄₇	C₂×Q₈	1	1	yes	characteristic
61	C₄²×C₄₇	#41	1	1	yes	characteristic
62	#10625329	—	4	2	no	—
63	#10625329	—	4	2	no	—
64	C₂×C₄₇⋊C₈	—	2	4	no	—
65	C₂×C₄₇⋊C₈	—	4	2	no	—
66	C₂³×C₄₇	C₂²⋊Q₈	1	1	yes	characteristic
67	C₂×C₄×C₄₇	—	2	2	no	—
68	C₂×C₄×C₄₇	—	2	4	no	—
69	C₂×C₄×C₄₇	—	4	1	no	—
70	C₂×C₄×C₄₇	—	4	2	no	—
71	C₂×C₄×C₄₇	—	4	2	no	—
72	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
73	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
74	C₂×C₄×C₄₇	C₄×Q₈	1	1	yes	characteristic
75	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
76	C₂×C₄×C₄₇	#125	1	1	yes	characteristic
77	C₂×C₄×C₄₇	C₄⋊Q₈	1	1	yes	characteristic
78	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
79	C₂×C₄×C₄₇	C₂×Q₁₆	1	1	yes	characteristic
80	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	1	yes	characteristic
81	C₂×C₄×C₄₇	#137	1	1	yes	characteristic
82	C₈×C₄₇	—	2	1	no	—
83	C₈×C₄₇	—	2	1	no	—
84	C₈×C₄₇	—	4	2	no	—
85	C₈×C₄₇	#107	1	2	yes	—
86	C₈×C₄₇	#107	1	2	yes	—
87	C₄₇⋊C₈	—	4	4	no	—
88	C₄₇⋊C₈	—	8	2	no	—
89	#9627	—	47	1	no	maximal, Sylow
90	C₂²×C₄₇	—	2	1	no	—
91	C₂²×C₄₇	—	2	1	no	—
92	C₂²×C₄₇	#389	1	1	yes	characteristic
93	C₂²×C₄₇	#474	1	1	yes	characteristic
94	C₂²×C₄₇	#477	1	1	yes	characteristic
95	C₄×C₄₇	—	2	1	no	—
96	C₄×C₄₇	—	2	1	no	—
97	C₄×C₄₇	—	2	1	no	—
98	C₄×C₄₇	—	4	4	no	—
99	C₄×C₄₇	C₂×#107	1	1	yes	characteristic
100	C₄×C₄₇	#428	1	1	yes	characteristic
101	C₄×C₄₇	#443	1	2	yes	—
102	C₄×C₄₇	#459	1	1	yes	characteristic
103	C₄×C₄₇	#462	1	1	yes	characteristic
104	#1848	—	47	1	no	—
105	C₈×C₄⋊C₄	D₄₇	1	1	yes	characteristic
106	#1571	—	47	2	no	—
107	#1564	—	47	1	no	—
108	#1303	—	47	2	no	—
109	C₂×C₄₇	—	2	1	no	—
110	C₂×C₄₇	—	2	1	no	—
111	C₂×C₄₇	#1945	1	1	yes	characteristic
112	C₂×C₄₇	#1947	1	1	yes	characteristic
113	C₂×C₄₇	#1953	1	1	yes	characteristic
114	#431	—	47	2	no	—
115	C₂×#108	—	94	1	no	—
116	C₂×#108	—	94	1	no	—
117	#423	—	47	2	no	—
118	C₂×C₄⋊C₈	C₂×D₄₇	1	1	yes	characteristic
119	C₂×C₄×C₈	—	2	1	no	—
120	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
121	C₄×C₄⋊C₄	C₂×D₄₇	1	1	yes	characteristic
122	#336	—	2	1	no	—
123	#335	—	47	2	no	—
124	#335	—	94	4	no	—
125	#334	—	47	2	no	—
126	C₄₇	#9627	1	1	yes	characteristic, complemented, Sylow
127	C₂×M₁₆	—	94	2	no	—
128	C₂²×C₈	—	2	1	no	—
129	C₂²×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
130	C₂²×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
131	C₂×C₄⋊C₄	—	2	1	no	—
132	C₂×C₄²	—	2	1	no	—
133	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
134	#108	—	188	2	no	—
135	#108	—	188	2	no	—
136	C₄⋊C₈	—	2	1	no	—
137	C₄⋊C₈	—	47	4	no	—
138	C₄⋊C₈	—	94	2	no	—
139	C₄⋊C₈	—	94	2	no	—
140	C₄⋊C₈	D₁₈₈	1	2	yes	—
141	C₂²⋊C₈	—	94	2	no	—
142	C₄×C₈	—	4	2	no	—
143	C₄×C₈	—	94	2	no	—
144	C₄×C₈	Dic₉₄	1	1	yes	characteristic
145	C₄×C₈	Dic₉₄	1	1	yes	characteristic
146	C₄×C₈	C₄×D₄₇	1	1	yes	characteristic
147	C₄×C₈	C₄×D₄₇	1	1	yes	characteristic
148	#95	Dic₉₄	1	2	yes	—
149	C₂²×C₄	—	2	2	no	—
150	C₂²×C₄	C₂×Dic₉₄	1	1	yes	characteristic
151	C₂²×C₄	#10625350	1	1	yes	characteristic
152	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
153	M₁₆	—	188	2	no	—
154	M₁₆	—	188	2	no	—
155	C₂×C₈	—	2	1	no	—
156	C₂×C₈	—	2	1	no	—
157	C₂×C₈	—	2	2	no	—
158	C₂×C₈	—	4	1	no	—
159	C₂×C₈	—	4	1	no	—
160	C₂×C₈	—	94	4	no	—
161	C₂×C₈	—	188	2	no	—
162	C₂×C₈	Dic₄₇⋊C₄	1	1	yes	characteristic
163	C₂×C₈	Dic₄₇⋊C₄	1	1	yes	characteristic
164	C₂×C₈	Dic₄₇⋊C₄	1	1	yes	characteristic
165	C₂×C₈	Dic₄₇⋊C₄	1	1	yes	characteristic
166	C₂×C₈	#10625334	1	2	yes	—
167	C₂×C₈	#10625337	1	2	yes	—
168	C₄⋊C₄	—	2	4	no	—
169	C₄²	—	4	1	no	—
170	C₄²	—	4	1	no	—
171	C₄²	C₂×Dic₉₄	1	1	yes	characteristic
172	C₄²	C₂×C₄×D₄₇	1	1	yes	characteristic
173	C₄²	C₂×D₁₈₈	1	1	yes	characteristic
174	C₄²	#10625350	1	1	yes	characteristic
175	C₂³	#13094003	1	1	yes	characteristic
176	C₂×C₄	—	2	2	no	—
177	C₂×C₄	—	2	4	no	—
178	C₂×C₄	—	4	1	no	—
179	C₂×C₄	—	4	2	no	—
180	C₂×C₄	—	4	2	no	—
181	C₂×C₄	C₂×Dic₄₇⋊C₄	1	1	yes	characteristic
182	C₂×C₄	C₄×D₁₈₈	1	1	yes	characteristic
183	C₂×C₄	C₄×Dic₉₄	1	1	yes	characteristic
184	C₂×C₄	#13093998	1	1	yes	characteristic
185	C₂×C₄	#13094016	1	1	yes	characteristic
186	C₂×C₄	#13094024	1	1	yes	characteristic
187	C₂×C₄	C₂×#10625334	1	1	yes	characteristic
188	C₂×C₄	C₂×#10625337	1	1	yes	characteristic
189	C₂×C₄	#13094049	1	1	yes	characteristic
190	C₂×C₄	#13094054	1	1	yes	characteristic
191	C₈	—	2	1	no	—
192	C₈	—	2	1	no	—
193	C₈	—	4	2	no	—
194	C₈	—	376	2	no	—
195	C₈	—	188	4	no	—
196	C₈	#13093957	1	2	yes	—
197	C₈	#13093957	1	2	yes	—
198	C₂²	—	2	1	no	—
199	C₂²	—	2	1	no	—
200	C₂²	#15689021	1	1	yes	characteristic
201	C₂²	#15689403	1	1	yes	characteristic
202	C₂²	#15689418	1	1	yes	characteristic
203	C₄	—	2	1	no	—
204	C₄	—	2	1	no	—
205	C₄	—	2	1	no	—
206	C₄	—	4	4	no	—
207	C₄	C₂×#13093957	1	1	yes	characteristic
208	C₄	#15689166	1	1	yes	characteristic
209	C₄	#15689241	1	2	yes	—
210	C₄	#15689318	1	1	yes	characteristic
211	C₄	#15689333	1	1	yes	characteristic
212	C₂	—	2	1	no	—
213	C₂	—	2	1	no	—
214	C₂	#20244136	1	1	yes	characteristic
215	C₂	#20244146	1	1	yes	characteristic
216	C₂	#20244178	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#13093998
commutator	C₄²×C₄₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₈×C₄⋊C₄×C₄₇	C₂
socle	C₂²×C₄₇	#389
prime radical	C₂³×C₄₇	C₂²⋊Q₈
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	C₈×C₄⋊C₄	D₄₇
hypocenter	C₄₇	#9627
LCS	G ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₈×C₄⋊C₄

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
10. #15688876	C₄	semidirect
12. #15688874	C₄	semidirect
126. C₄₇	#9627	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#20243560	2
#20242162	2
#20242136	2
#20241291	2
C₈×C₄⋊C₄×C₄₇	2
#9627	47